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MAT F 3 LESSON 1

quadratic equations

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MAT F 3 LESSON 1

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  1. Topic 1: Quadratic Equations FORM 3 MATHEMATICS BY: B.E Namithambo, BEd.Sc-Chemistry 07/04/2025

  2. Introduction to Quadratic Equations • The term "quadratic" comes from the Latin word "quadratus," which means "square ❑Squaring something is like we are raising it to power 2. -Quadratic equations are algebraic expressions of the form ax² + bx + c = 0. wherea ≠0 . Key terms: 1. Variable: A symbol representing an unknown value (e.g., x, y, t, m, n, q etc). 2. Coefficient: A number multiplying a variable (e.g., in 3x, 3 is the coefficient, in y 1 is a coefficient, in 1/3m, 1/3 is a coefficient , ax² + bx + c , a and b are coefficients). 3. Constant: A fixed number (e.g., in x + 4, 4 is a constant in ax² + bx + c, c is a constant).

  3. Introduction cont… 4. Term: A part of the expression separated by + or -.( e.g., in 2x+ 1, we have two terms, which are 2x and 1, y-4x+ m, we have three terms, which are y, -4x and m , and in ax² + bx + c , we have three terms, ax², bx and c 5.Degree: The highest power of the variable (degree of x² + x+3 is 2). • In quadrat equation, the degree is 2 • So, any algebraic expression with a degree 2 is a quadratic expression. 6 binomial: any algebraic expression with exactly two unlike terms separated by + or –

  4. Factori Factoris sing ing Quadratic Expressions Quadratic Expressions Quadratic expression are algebraic expressions of the form ax² + bx + c, a ≠0 . Factorisation is the process of breaking an expression into its factors. • The degree of quadratic expression is 2 General method for factorsing quadratic expression Step 1. Multiply coefficient of x² and constant(c) Step 2: Find two numbers that: •Multiply to give the product from step 1 •Add up to the coefficient of x Step 3: Rewrite the expression by splitting the middle term (bx) into two terms using the two numbers found in Step 2. Step 4: Group the terms into two pairs. Step 5: Factor out the greatest common factor (GCF) from each pair. Step 6: Factor out the common binomial factor. Step 7: Write the expression as a product of two binomials

  5. Factorisation cont.. Factorisation cont.. Example 1: Factorise ?2+5x+6 step 1: Multiply the coefficient of ?2(which is 1) and the constant term (which is 6). 1×6 = 6 Step 2: Find two numbers that multiply to give 6( the product of step 1) and add up to 5 (the coefficient of x). The two numbers are 2 and 3, because: 2×3=6 and 2+3=5

  6. Example 1 cont… Step 3: Rewrite the expression b splitting the middle term (5x) into two terms using the numbers found in Step 2. x² + 2x+3x + 6 Step 4: Group the terms in pairs. (x²+2x)+(3x+6)

  7. Example 1 cont… Step 5: Factor each pair separately. • For (x²+2x) factor out x, giving x( x +2x) • For (3x+6) factor out 3, giving 3(x+2) x(x+2)+3(x+2) Step 6: Now, factor out the common binomial factor (x+2). (x+2)+(x+3)

  8. Solving Example 1 in one step Factorise ?2+5x+6 ?2+5x+6 6 ?2 ?2 +2x+3x+6 x(x+2)+3(x+2) (x+1)(x+2)

  9. Factorisation cont.. Factorisation cont.. Example 2: Factorise ?2−5x+6 step 1: Multiply the coefficient of ?2 (which is 1) and the constant term (which is 6). 1×6 = 6 Step 2: Find two numbers whose product is 6 and whose sum is -5 The two numbers are 2 and 3, because: -2×-3=6 and -2+(-3)= -5

  10. Example 2 cont… Step 3:Rewrite the expression by splitting the middle term using using -2 and -3 x² -2x-3x + 6 Step 4:Group the terms (x² -2x)-(3x-6)

  11. Example 2 cont… Step 5: Factor each group. x(x-2)-3(x-2) Step 6: Now, factor out the common binomial (x-2)(x-3) So, the factorized form of ?2−5x+6 is (x-2)(x-3)

  12. Solving Example 2 in one step Factorise ?2−5x+6 ?2+5x+6 6 ?2 ?2 -2x-3x+6 x(x-2)-3(x-2) (x-1)(x-2)

  13. Practice exercise Factorise following quadratic expressions: 1. 6?2+11x−35 2. ?2−7x+10 3. ?2−3x−10 4. 3?2+5x−12 5. 2?2+7x+3

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